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SageMath
E = EllipticCurve("ed1")
E.isogeny_class()
Elliptic curves in class 121275ed
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121275.fg3 | 121275ed1 | \([1, -1, 0, -179073792, -922306095509]\) | \(473897054735271721/779625\) | \(1044772063822265625\) | \([2]\) | \(10616832\) | \(3.1493\) | \(\Gamma_0(N)\)-optimal |
| 121275.fg2 | 121275ed2 | \([1, -1, 0, -179128917, -921709808384]\) | \(474334834335054841/607815140625\) | \(814530420257433837890625\) | \([2, 2]\) | \(21233664\) | \(3.4959\) | |
| 121275.fg4 | 121275ed3 | \([1, -1, 0, -130894542, -1429280136509]\) | \(-185077034913624841/551466161890875\) | \(-739017399502162094466796875\) | \([2]\) | \(42467328\) | \(3.8425\) | |
| 121275.fg1 | 121275ed4 | \([1, -1, 0, -228245292, -375977765759]\) | \(981281029968144361/522287841796875\) | \(699915659943431854248046875\) | \([2]\) | \(42467328\) | \(3.8425\) |
Rank
sage: E.rank()
The elliptic curves in class 121275ed have rank \(0\).
Complex multiplication
The elliptic curves in class 121275ed do not have complex multiplication.Modular form 121275.2.a.ed
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
